https://www.desmos.com/calculator/s6qdwrlqye

These were the equations I used throughout my graphing process:

Quadratic and reciprocal functions were used because of the slight curve they provided. A cubic equation was used on the weird occasion regarding the curves in his pants, to get a softer curve. These were specific to the outline of the bodies, mostly used because they were easy to shape. When moving these equations, I would place the variables within brackets, subtracting or adding numbers from the y variable in order to move them vertically. When moving them horizontally, I subtracted or added outside of the squared equation, essentially changing the overall value, moving it farther down the x axis.

The circle equations were used for objects like Patrick’s eyeballs and stomach. To make the circle smaller or larger, I input numbers into the r variable. In order to move them vertically and horizontally, I did the same thing, placing the variables within brackets and subtracting or adding.

Linear equations were used for the straight line which were simpler. I placed coefficients in front of the x variable in order to change his angle, subtracting or adding to change the x axis like the other equations.

I didn’t face any real challenges, other than wanting to work around images and instead go off of what I could see. As for “aha moments,” there was one early on, when working with the placement of a circle. When creating Patrick’s eyes, I had no clue as to how I would change the circles x and y, specifically because I had this idea that the equation, r^{2} = x^{2} + y^{2}, could not be altered. After realizing that I could place the variables in brackets and alter from there, it got much easier.

I didn’t require much help throughout the process, although when needed I did ask my table group. As for techniques, I placed each individual equation out as a model, looking between the picture and the graph to see what I could apply. This worked very well. Overall, this assignment allowed me to better understand how to alter the location and size of a graph relative to the variables placed into the equation, like subtracting or adding from the x, y, or overall value. Not only the size and location, but the distortion as well. It allowed for a better understanding of how manipulating equations really work, as well as assisting with my knowledge of domain and range.